It means that they would exhibit Reepicheep at fairs as a side-show. Because in the human world talking mice would be unusual he would be something used as a freak in a sideshow. As for the actual grammar, show in this case means the same as exhibit, they would "show him at fairs". It is the same as if I said "I will show you this book". If you need any more explanation feel free to ask. By the way that is a great book so I'm happy to see someone reading it.